Algebra Examples

$- 1 + 9 x < 2 + 3 (- 9 x + \frac{1}{2})$

Step 1

Simplify $2 + 3 (- 9 x + \frac{1}{2})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- 1 + 9 x < 2 + 3 (- 9 x) + 3 (\frac{1}{2})$

Step 1.1.2

Multiply $- 9$ by $3$ .

$- 1 + 9 x < 2 - 27 x + 3 (\frac{1}{2})$

Step 1.1.3

Combine $3$ and $\frac{1}{2}$ .

$- 1 + 9 x < 2 - 27 x + \frac{3}{2}$

Step 1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 1 + 9 x < - 27 x + 2 \cdot \frac{2}{2} + \frac{3}{2}$

Step 1.3

Combine $2$ and $\frac{2}{2}$ .

$- 1 + 9 x < - 27 x + \frac{2 \cdot 2}{2} + \frac{3}{2}$

Step 1.4

Combine the numerators over the common denominator.

$- 1 + 9 x < - 27 x + \frac{2 \cdot 2 + 3}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $2$ by $2$ .

$- 1 + 9 x < - 27 x + \frac{4 + 3}{2}$

Step 1.5.2

Add $4$ and $3$ .

$- 1 + 9 x < - 27 x + \frac{7}{2}$

Step 2

Move all terms containing $x$ to the left side of the inequality.

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Step 2.1

Add $27 x$ to both sides of the inequality.

$- 1 + 9 x + 27 x < \frac{7}{2}$

Step 2.2

Add $9 x$ and $27 x$ .

$- 1 + 36 x < \frac{7}{2}$

Step 3

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1

Add $1$ to both sides of the inequality.

$36 x < \frac{7}{2} + 1$

Step 3.2

Write $1$ as a fraction with a common denominator.

$36 x < \frac{7}{2} + \frac{2}{2}$

Step 3.3

Combine the numerators over the common denominator.

$36 x < \frac{7 + 2}{2}$

Step 3.4

Add $7$ and $2$ .

$36 x < \frac{9}{2}$

Step 4

Divide each term in $36 x < \frac{9}{2}$ by $36$ and simplify.

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Step 4.1

Divide each term in $36 x < \frac{9}{2}$ by $36$ .

$\frac{36 x}{36} < \frac{\frac{9}{2}}{36}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $36$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{36 x}{36} < \frac{\frac{9}{2}}{36}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x < \frac{\frac{9}{2}}{36}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x < \frac{9}{2} \cdot \frac{1}{36}$

Step 4.3.2

Cancel the common factor of $9$ .

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Step 4.3.2.1

Factor $9$ out of $36$ .

$x < \frac{9}{2} \cdot \frac{1}{9 (4)}$

Step 4.3.2.2

Cancel the common factor.

$x < \frac{9}{2} \cdot \frac{1}{9 \cdot 4}$

Step 4.3.2.3

Rewrite the expression.

$x < \frac{1}{2} \cdot \frac{1}{4}$

Step 4.3.3

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$x < \frac{1}{2 \cdot 4}$

Step 4.3.4

Multiply $2$ by $4$ .

$x < \frac{1}{8}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{1}{8}$

Interval Notation:

$(- \infty, \frac{1}{8})$